Expectation minimum—a new principle of inverse problem theory in the photothermal sciences: theoretical characterization of expectation values

Abstract

The expectation minimum ({EM)} principle is a new method of inverse problem theory that can be used to obtain stable solutions to first-kind Fredholm integrals. The {EM} principle uses the addition of noise to a basis of model vectors that are related to the experimental data by a multilinear discrete model. The experimental data are projected onto this randomized basis set and solution vectors for individual data projections are recovered by any one of a number of standard algorithms. The solutions are averaged arithmetically to obtain a stable estimate of the minimum error solution by expectation. It is shown from theory that the presence of noise in the model basis provides a method of regularization of the problem. Examples applicable to the problem of source localization in inverse heat conduction are examined. When factorization methods are used to recover the solution vectors corresponding to each data projection, the solutions recovered by the {EM} principle are shown to quantitatively converge to the zero-order uniform Tikhonov regularized solutions. When a stepwise projection method is used in lieu of factorizations, the recovered solutions agree nearly quantitatively with the solutions recovered using the autocorrelation method of linear filtering derived via a modified form of the Yule-Walker equations, The resolution and robustness of the {EM} solutions in this second case are greatly superior to those obtained by the autocorrelation method. The ultimate accuracy and resolution of the {EM} solutions in this latter case is limited by the noise in the model basis. Through appropriate sampling methods, the noise in the model basis may be set as low as three orders of magnitude below the noise on the experimental data, This gives an enormous improvement in performance over the previous methods of solution using regularization. (C) 1997 Society of Photo-Optical instrumentation Engineers.